Conformal structure of massless scalar amplitudes beyond tree level

  • Nabamita Banerjee
  • Shamik Banerjee
  • Sayali Atul Bhatkar
  • Sachin Jain
Open Access
Regular Article - Theoretical Physics
  • 8 Downloads

Abstract

We show that the one-loop on-shell four-point scattering amplitude of massless ϕ4 scalar field theory in 4D Minkowski space time, when Mellin transformed to the Celestial sphere at infinity, transforms covariantly under the global conformal group (SL(2, ℂ)) on the sphere. The unitarity of the four-point scalar amplitudes is recast into this Mellin basis. We show that the same conformal structure also appears for the two-loop Mellin amplitude. Finally we comment on some universal structure for all loop four-point Mellin amplitudes specific to this theory.

Keywords

Conformal Field Theory Global Symmetries Scattering Amplitudes 

Notes

Open Access

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References

  1. [1]
    P.A.M. Dirac, Wave equations in conformal space, Annals Math. 37 (1936) 429 [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    J. de Boer and S.N. Solodukhin, A Holographic reduction of Minkowski space-time, Nucl. Phys. B 665 (2003) 545 [hep-th/0303006] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  3. [3]
    T. Banks, A Critique of pure string theory: Heterodox opinions of diverse dimensions, hep-th/0306074 [INSPIRE].
  4. [4]
    G. Barnich and C. Troessaert, Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited, Phys. Rev. Lett. 105 (2010) 111103 [arXiv:0909.2617] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    G. Barnich and C. Troessaert, Supertranslations call for superrotations, PoS(CNCFG2010)010 [Ann. U. Craiova Phys. 21 (2011) S11] [arXiv:1102.4632] [INSPIRE].
  6. [6]
    D. Kapec, V. Lysov, S. Pasterski and A. Strominger, Semiclassical Virasoro symmetry of the quantum gravity S-matrix, JHEP 08 (2014) 058 [arXiv:1406.3312] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    D. Kapec, P. Mitra, A.-M. Raclariu and A. Strominger, 2D Stress Tensor for 4D Gravity, Phys. Rev. Lett. 119 (2017) 121601 [arXiv:1609.00282] [INSPIRE].
  8. [8]
    C. Cheung, A. de la Fuente and R. Sundrum, 4D scattering amplitudes and asymptotic symmetries from 2D CFT, JHEP 01 (2017) 112 [arXiv:1609.00732] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    S. He, Y.-t. Huang and C. Wen, Loop Corrections to Soft Theorems in Gauge Theories and Gravity, JHEP 12 (2014) 115 [arXiv:1405.1410] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    M. Bianchi, S. He, Y.-t. Huang and C. Wen, More on Soft Theorems: Trees, Loops and Strings, Phys. Rev. D 92 (2015) 065022 [arXiv:1406.5155] [INSPIRE].
  11. [11]
    Z. Bern, S. Davies and J. Nohle, On Loop Corrections to Subleading Soft Behavior of Gluons and Gravitons, Phys. Rev. D 90 (2014) 085015 [arXiv:1405.1015] [INSPIRE].
  12. [12]
    Z. Bern, S. Davies, P. Di Vecchia and J. Nohle, Low-Energy Behavior of Gluons and Gravitons from Gauge Invariance, Phys. Rev. D 90 (2014) 084035 [arXiv:1406.6987] [INSPIRE].
  13. [13]
    S.W. Hawking, M.J. Perry and A. Strominger, Superrotation Charge and Supertranslation Hair on Black Holes, JHEP 05 (2017) 161 [arXiv:1611.09175] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    F. Cachazo and A. Strominger, Evidence for a New Soft Graviton Theorem, arXiv:1404.4091 [INSPIRE].
  15. [15]
    A. Strominger, Asymptotic Symmetries of Yang-Mills Theory, JHEP 07 (2014) 151 [arXiv:1308.0589] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    S. Pasterski, S.-H. Shao and A. Strominger, Flat Space Amplitudes and Conformal Symmetry of the Celestial Sphere, Phys. Rev. D 96 (2017) 065026 [arXiv:1701.00049] [INSPIRE].
  17. [17]
    S. Pasterski and S.-H. Shao, Conformal basis for flat space amplitudes, Phys. Rev. D 96 (2017) 065022 [arXiv:1705.01027] [INSPIRE].
  18. [18]
    S. Pasterski, S.-H. Shao and A. Strominger, Gluon Amplitudes as 2d Conformal Correlators, Phys. Rev. D 96 (2017) 085006 [arXiv:1706.03917] [INSPIRE].
  19. [19]
    C. Cardona and Y.-t. Huang, S-matrix singularities and CFT correlation functions, JHEP 08 (2017) 133 [arXiv:1702.03283] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    H.T. Lam and S.-H. Shao, Conformal Basis, Optical Theorem and the Bulk Point Singularity, arXiv:1711.06138 [INSPIRE].
  21. [21]
    P. Paul, Conformal structure of QED Amplitudes, in progress.Google Scholar
  22. [22]
    P. Agrawal and R. Sundrum, Small Vacuum Energy from Small Equivalence Violation in Scalar Gravity, JHEP 05 (2017) 144 [arXiv:1611.07021] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    H. Kleinert and V. Schulte-Frohlinde, Critical properties of ϕ 4 -theories, World Scientific, Singapore (2004) [INSPIRE].

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Nabamita Banerjee
    • 1
  • Shamik Banerjee
    • 2
  • Sayali Atul Bhatkar
    • 1
  • Sachin Jain
    • 1
  1. 1.Indian Institute of Science Education and ResearchPuneIndia
  2. 2.Institute of PhysicsBhubaneswarIndia

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